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Rank (graph Theory)
In graph theory, a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let equal the number of vertices of the graph. * In the matrix theory of graphs the rank of an undirected graph is defined as the rank of its adjacency matrix. :Analogously, the nullity of the graph is the nullity of its adjacency matrix, which equals . * In the matroid theory of graphs the rank of an undirected graph is defined as the number , where is the number of connected components of the graph. Equivalently, the rank of a graph is the rank of the oriented incidence matrix associated with the graph.. See in particular the discussion on p. 218. :Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula , where ''n'' and ''c'' are as above and ''m'' is the number of edges in the graph. The nullity is equal to the first Betti number of the graph. The sum of the rank and the nullity is the number of edges. Examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element of ''X'' and one column for each element of ''Y''. The entry in row ''x'' and column ''y'' is 1 if ''x'' and ''y'' are related (called ''incident'' in this context) and 0 if they are not. There are variations; see below. Graph theory Incidence matrix is a common graph representation in graph theory. It is different to an adjacency matrix, which encodes the relation of vertex-vertex pairs. Undirected and directed graphs In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented. The ''unoriented incidence matrix'' (or simply ''incidence matrix'') of an undirected graph is a n\times m matrix ''B'', where ''n'' and ''m'' are the numbers of vertices and edges respectively, such that :B_=\left\{\begin{a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Graph Theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. Branches of algebraic graph theory Using linear algebra The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter ''D'' w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Rank
In graph theory, the cycle rank of a directed graph is a digraph connectivity measure proposed first by Eggan and Büchi . Intuitively, this concept measures how close a digraph is to a directed acyclic graph (DAG), in the sense that a DAG has cycle rank zero, while a complete digraph of order ''n'' with a self-loop at each vertex has cycle rank ''n''. The cycle rank of a directed graph is closely related to the tree-depth of an undirected graph and to the star height of a regular language. It has also found use in sparse matrix computations (see ) and logic . Definition The cycle rank ''r''(''G'') of a digraph is inductively defined as follows: * If ''G'' is acyclic, then . * If ''G'' is strongly connected and ''E'' is nonempty, then ::r(G) = 1 + \min_ r(G-v),\, where is the digraph resulting from deletion of vertex and all edges beginning or ending at . * If ''G'' is not strongly connected, then ''r''(''G'') is equal to the maximum cycle rank among all strongly connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circuit Rank
In graph theory, a branch of mathematics, the circuit rank, cyclomatic number, cycle rank, or nullity of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. It is equal to the number of independent cycles in the graph (the size of a cycle basis). Unlike the corresponding feedback arc set problem for directed graphs, the circuit rank is easily computed using the formula :r = m - n + c, where is the number of edges in the given graph, is the number of vertices, and is the number of connected components. . It is also possible to construct a minimum-size set of edges that breaks all cycles efficiently, either using a greedy algorithm or by complementing a spanning forest. The circuit rank can be explained in terms of algebraic graph theory as the dimension of the cycle space of a graph, in terms of matroid theory as the corank of a graphic matroid, and in terms of topology as one of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betti Number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if H_n(X) \cong 0 then b_n(X) = 0, if H_n(X) \cong \mathbb then b_n(X) = 1, if H_n(X) \cong \mathbb \oplus \mathbb then b_n(X) = 2, if H_n(X) \cong \mathbb \oplus \mathbb\oplus \mathbb then b_n(X) = 3, etc. Note that only the ranks of infinite groups are considered, so for example if H_n(X) \cong \mathbb^k \oplus \mathbb/(2) , where \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (matrix)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: :\ker(L) = \left\ . Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows that the image of is isomorphic to the quotient of by the ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank (linear Algebra)
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental result in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undirected Graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Component (graph Theory)
In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not. The components of a graph can be constructed in linear time, and a special case of the problem, connected-component labeling, is a basic technique in image analysis. Dynamic co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Theory
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawski' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
